U.S. Secretary of State Mike Pompeo, third from left, South Korean Foreign Minister Kang Kyung-wha, center, and Japanese Foreign Minister Taro Kono, second from right, attend their meeting at Foreign Ministry in Seoul, South Korea, Thursday, June14, 2018. (Kim Hong-ji/Pool Photo via AP)

U.S. Secretary of State Mike Pompeo, center, South Korean Foreign Minister Kang Kyung-wha, right, and Japanese Foreign Minister Taro Kono leave after a joint press conference following their meeting at Foreign Ministry in Seoul, South Korea, Thursday, June 14, 2018. (AP Photo/Ahn Young-joon)

Japanese Foreign Minister Taro Kono, left, speaks as U.S. Secretary of State Mike Pompeo, right, and South Korean Foreign Minister Kang Kyung-wha listen during a joint press conference following their meeting at Foreign Ministry in Seoul, South Korea, Thursday, June 14, 2018. (AP Photo/Ahn Young-joon)

U.S. Secretary of State Mike Pompeo, right, and Japanese Foreign Minister Taro Kono, left, listen to South Korean Foreign Minister Kang Kyung-wha during a joint press conference following their meeting at Foreign Ministry in Seoul, South Korea, Thursday, June 14, 2018. (AP Photo/Ahn Young-joon)

U.S. Secretary of State Mike Pompeo, right, speaks as South Korean Foreign Minister Kang Kyung-wha, center, and Japanese Foreign Minister Taro Kono listen during a joint press conference following their meeting at Foreign Ministry in Seoul, South Korea, Thursday, June 14, 2018. (AP Photo/Ahn Young-joon)

WASHINGTON (AP) — America and the world can "sleep well tonight," President Donald Trump declared on Wednesday, boasting that his summit with Kim Jong Un had ended any nuclear threat from North Korea though the meeting produced no details on how or when weapons might be eliminated or even reduced.

While Trump claimed a historic breakthrough at the most significant diplomatic event of his presidency, Secretary of State Mike Pompeo, was more measured. He said that Trump's tweet was made "with eyes wide open" to the possibility diplomacy could falter, and that the U.S. wants North Korea to take "major" nuclear disarmament steps within the next two years — before the end of Trump's first term in 2021.

And while North Korean state media had claimed that Trump and Kim agreed to "step-by-step" actions — an apparent euphemism for phased sanctions relief in exchange for phased denuclearization — Pompeo ruled that out. He insisted that Trump had been explicit about the sequencing from the start.

"We're going to get denuclearization," Pompeo said. "Only then will there be relief from the sanctions."

Shortly after arriving in Seoul to brief U.S. treaty allies Japan and South Korea, Pompeo also cautioned that the U.S. would resume "war games" with close ally South Korea if the North stops negotiating in good faith.

The president had announced a halt in the drills after his meeting with Kim on Tuesday, a concession long sought by Pyongyang but generally opposed by Seoul and Tokyo. After a three-way meeting with Pompeo and Japan's top diplomat, South Korean Foreign Minister Kang Kyung-wha suggested the U.S. still had some explaining to do, telling reporters that the issue of the drills "was not discussed in depth."

"This is a matter that military officials from South Korea and the United States will have to discuss further and coordinate," Kang said in Korean.

The summit in Singapore did mark a reduction in tensions — a sea change from last fall, when North Korea was conducting nuclear and missile tests and Trump and Kim were trading threats and insults that stoked fears of war. Kim is now promising to work toward a denuclearized Korean Peninsula.

But the details of what is sure to be a complex and contentious process have yet to be settled.

Despite the uncertainties, Trump talked up the outcome of what was the first meeting between a U.S. and North Korean leader in six decades of hostility. The Korean War ended in 1953 without a peace treaty, leaving the two sides in a technical state of war.

"Just landed - a long trip, but everybody can now feel much safer than the day I took office," Trump tweeted early Wednesday. "There is no longer a Nuclear Threat from North Korea. Meeting with Kim Jong Un was an interesting and very positive experience. North Korea has great potential for the future!"

Pompeo said the brief, four-point joint statement that emerged from the summit did not encapsulate all the progress the U.S. and North Korea had made. He said negotiations would recommence "in the next week or so."

He bristled at questions from reporters about the vague wording of the statement where North Korea "commits to work toward complete denuclearization of the Korean Peninsula" — a promise it has made several times before in the past 25 years and reneged on. Although Trump before the summit had insisted on "complete, verifiable and irreversible" denuclearization, the deal he signed with Kim made no mention of those last two conditions.

Irrelevant, Pompeo argued Thursday in Seoul, noting that because the deal makes reference to a previous agreement that did include verification, it automatically "incorporates" verification without having to state it outright.

On Thursday, the rival Koreas held rare high-level military talks to discuss reducing tensions across their heavily fortified border. It's possible North Korean officials will seek a firm commitment from the South on stopping its military drills with the United States.

Seoul's Defense Ministry said the talks would focus on carrying out agreements from a summit between Kim and South Korean President Moon Jae-in where they vowed to take materialized steps to reduce military tensions and eliminate the danger of war.

In public, at least, South Korea's leader sought to put a positive spin on the summit's outcome. Meeting briefly with Pompeo on Thursday morning, Moon said he was eager to hear how the United States and North Korea could "fully and expeditiously implement this great agreement."

"The summit was a truly historic feat which moved us from the era of hostility towards the era of dialogue, of peace and prosperity," Moon said through a translator

Pompeo, the former CIA director, planned to fly to Beijing later Thursday to update the Chinese government about the talks.

While Trump was facing questions at home and among allies about whether he gave away too much in return for too little, North Korean state media heralded claims of a victorious meeting with the U.S. president. Photos of Kim standing side-by-side with Trump on the world stage were splashed across newspapers.

Trump's own chest-thumping tweet seemed reminiscent of the "Mission Accomplished" banner flown behind President George W. Bush in 2003 when he spoke aboard a Navy ship following the U.S. invasion of Iraq. The words came back to haunt the administration, as the war dragged on throughout Bush's presidency.

Independent experts say the North could have enough fissile material for between about a dozen and 60 nuclear bombs. Last year it tested long-range missiles that could reach the U.S. mainland, although it remains unclear if it has mastered the technology to deliver a nuclear warhead that could re-enter the atmosphere and hit its target.

"Before taking office people were assuming that we were going to War with North Korea," Trump tweeted. "President (Barack) Obama said that North Korea was our biggest and most dangerous problem. No longer - sleep well tonight!"

Actually, concerns about North Korean missiles and nuclear weapons reached a peak last year, during Trump's first year in office, as the North conducted more tests and Trump and Kim aimed ever more fiery rhetoric at each other.

Christopher Hill, chief U.S. negotiator with North Korea in the George W. Bush administration, suggested in an interview that it's "a little premature" for Trump to say Kim is someone the U.S. can trust.

"Kim Jong Un has proved to be a pretty ruthless leader in North Korea, and I'm not sure this sort of speed dating of a 45-minute one-on-one meeting ... would suggest that there's nothing to be concerned about," he said.

Freezing the regular military exercises with South Korea is a major concession to North Korea that has long claimed the drills were invasion preparations. Trump's announcement appeared to catch the Pentagon and officials in Seoul off guard, and some South Koreans were alarmed. Trump cast the decision as a cost-saving measure, but also called the exercises "inappropriate" while talks continue.

Pompeo, speaking to reporters just after he arrived in Seoul, said he was there when Trump talked about it with Kim, and the president "made very clear" that the condition for the freeze was that good-faith talks be ongoing. He told reporters that if the U.S. concludes they no longer are, the freeze "will no longer be in effect."

"He was unambiguous about that," Pompeo said.

In North Korea on Wednesday, Pyongyang's first reports on the summit stressed to the nation's people that Trump had agreed to Kim's demand to halt the military exercises and suggested that Trump also said he would lift sanctions as negations progressed.

"President Trump appreciated that an atmosphere of peace and stability was created on the Korean Peninsula and in the region, although distressed with the extreme danger of armed clash only a few months ago, thanks to the proactive peace-loving measures taken by the respected Supreme Leader from the outset of this year," the North's state-run Korean Central News Agency said in a summary of the meeting.

In Japan, the prospect of canceled U.S.-South Korean drills was met with concern.

"The U.S.-South Korea joint exercises and U.S. forces in South Korea play significant roles for the security in East Asia," Defense Minister Itsunori Onodera told reporters Wednesday. He said he planned to continue sharing the view with Washington and Seoul.

The U.S. has stationed combat troops in South Korea since the end of the Korean War and has used them in a variety of drills. The next scheduled major exercise, involving tens of thousands of troops, normally would be held in August.